Statutory birth registration started in Ireland in 1864. Prior to that you need to rely on church records, where they exist. To do that you need to know the person’s religious denomination. Do you know what the Cox & Cluff families were? Looking at the 1901 census of Fermanagh, nearly all the Cox were RC, and all the Cluff were Church of Ireland (Anglican).
If the family were RC, then there were 2 churches in the parish at that time. Enniskillen has baptisms from 1838 and marriages from 1817; Tempo has baptisms and marriages from 1845. The Church of Ireland has almost no records for Enniskillen parish before 1861. (They were lost in the 1922 fire in Dublin). Tempo has baptisms from 1836 onwards. The RC parish records are on-line free on the National Library site:
There’s a copy of the Church of Ireland records for Tempo in PRONI (public record office) in Belfast. They are not on-line anywhere so far as I am aware.
I had a look for a marriage between John & Jane. RC marriages weren’t recorded in the statutory records prior to 1864. Other denominations and register office ceremonies were recorded from 1845 onwards. I didn’t find a marriage for them. That suggests to me that, if they married in Ireland, Jane converted to RC and they had an RC marriage. If so you might find an adult baptism for her, typically a month or so before the marriage. You can search for both the baptism and the marriage in the RC records on the nli site.